scholarly journals Analysis of a Fractional-order “SVEIR” Epidemic Mo del with a General Nonlinear Saturated Incidence Rate in a Continuous Reactor

2019 ◽  
pp. 1-17
Author(s):  
Miled El Hajji ◽  
Sayed Sayari
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
System Modeling ◽  
Equilibrium Points ◽  
Continuous Reactor ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Dimensional Dynamical System

In this paper, I propose a fractional-order mathematical five-dimensional dynamical system  modeling a SVEIR model of infectious disease transmission in a chemostat. A profound qualitative analysis is given. The analysis of the local and global stability of equilibrium points is carried out. It is proved that if R > 1, then the disease-persistence (endemic) equilibrium is globally asymptotically stable. However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable in R 5. Finally, some numerical tests are done using the ”PECE” method in order to validate the obtained results.

scholarly journals Mathematical Analysis of a Fractional-order "SIR" Epidemic Model with a General Nonlinear Saturated Incidence Rate in a Chemostat

2019 ◽  
pp. 1-12
Author(s):  
Miled El Hajji
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Continuous Reactor ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Saturated Incidence Rate ◽  
Saturated Incidence

In the present work, a fractional-order differential equation based on the Susceptible-Infected- Recovered (SIR) model with nonlinear incidence rate in a continuous reactor is proposed. A profound qualitative analysis is given. The analysis of the local and global stability of equilibrium points is carried out. It is proved that if the basic reproduction number R > 1 then the disease-persistence (endemic) equilibrium is globally asymptotically stable. However, if R ≤ 1, then the disease-free equilibrium is globally asymptotically stable. Finally, some numerical tests are done in order to validate the obtained results.

scholarly journals Mathematical study for the phase-based transmissibility of a novel COVID-19 Coronavirus

2020 ◽  
Author(s):  
Miled EL HAJJI ◽  
Sayed SAYARI ◽  
Abdelhamid ZAGHDANI
Keyword(s):  
Disease Transmission ◽  
System Modeling ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Mathematical Study ◽  
Infectious Disease Transmission ◽  
Globally Asymptotically Stable ◽  
Dynamical System Modeling ◽  
Disease Free ◽  
Disease Persistence

Abstract In this paper, a mathematical dynamical system modeling a SEIRW model of infectious disease transmission for a transmissibility of a novel COVID-19 Coronavirus is studied. A qualitative analysis such as the local and global stability of equilibrium points is carried out.It is proved that if $\R \leq 1$, then the disease-free equilibrium is globally asymptotically stable and if $\R > 1$, then the disease-persistence equilibrium is globally asymptotically stable.

scholarly journals Global Stability and Optimal Control of Dengue with Two Coexisting Virus Serotypes

MATEMATIKA ◽  
2019 ◽  
Vol 35 (4) ◽  
pp. 149-170
Author(s):  
Afeez Abidemi ◽  
Rohanin Ahmad ◽  
Nur Arina Bazilah Aziz
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Boundary Equilibrium

This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-freeequilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j 2 {2, 3, 4}), respectively, satisfy R01,R0j 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1, j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.

scholarly journals Mathematical Analysis of TB Model with Vaccination and Saturated Incidence Rate

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Ashenafi Kelemu Mengistu ◽  
Peter J. Witbooi
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Model System ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Latently Infected ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free ◽  
The Basic Reproduction Number

The model system of ordinary differential equations considers two classes of latently infected individuals, with different risk of becoming infectious. The system has positive solutions. By constructing a Lyapunov function, it is proved that if the basic reproduction number is less than unity, then the disease-free equilibrium point is globally asymptotically stable. The Routh-Hurwitz criterion is used to prove the local stability of the endemic equilibrium when R 0 > 1 . The model is illustrated using parameters applicable to Ethiopia. A variety of numerical simulations are carried out to illustrate our main results.

scholarly journals Analisis Kestabilan Model Matematika Penyebaran Penyakit Schistosomiasis dengan Saturated Incidence Rate

2020 ◽  
Vol 2 (2) ◽  
pp. 71
Author(s):  
Elda Widya ◽  
Miswanto Miswanto ◽  
Cicik Alfiniyah
Keyword(s):  
Equilibrium Point ◽  
Incidence Rate ◽  
Infected Snail ◽  
Equilibrium Points ◽  
Incidence Rates ◽  
Endemic Equilibrium ◽  
Human Populations ◽  
Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence

Schistosomiasis is a disease caused by infections of the genus Schistosoma. Schistosomiasis can be transmitted through schistosoma worms that contact human skin. Schistosomiasis is a disease that continues to increase in spread. Saturated incidence rates pay attention to the ability to infect a disease that is limited by an increase in the infected population. This thesis formulates and analyzes a mathematical model of the distribution of schistosomiasis with a saturated incidence rate. Based on the analysis of the model, two equilibrium points are obtained, namely non-endemic equilibrium points (E0) and endemic equilibrium points (E1). Both equilibrium points are conditional asymptotically stable. The nonendemic equilibrium point will be asymptotically stable if rh > dh, rs > ds and R0 < 1, while the endemic equilibrium point will be asymptotically stable if R0 > 1. Sensitivity analysis shows that there are parameters that affect the spread of the disease. Based on numerical simulation results show that when R0 < 1, the number of infected human populations (Hi), the number of infected snail populations (Si), the amount of cercaria density (C) and the amount of miracidia density (M) will tend to decrease until finally extinct. Otherwise at the time R0 > 1, the number of the four populations tends to increase before finally being in a constant state.

scholarly journals Dynamical analysis of a fractional order model incorporating fear in the disease transmission rate of COVID-19

2020 ◽  
Vol 99 (99) ◽  
pp. 1-17
Author(s):  
Debasis Mukherjee Debasis Mukherjee ◽  
Chandan Maji
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
Transmission Rate ◽  
Three Dimensional ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Fear Effect ◽  
And Control ◽  
Disease Free

This paper deals with a fractional-order three-dimensional compartmental model with fear effect. We have investigated whether fear can play an important role or not to spread and control the infectious diseases like COVID-19, SARS etc. in a bounded region. The basic results on uniqueness, non-negativity and boundedness of the solution of the system are investigated. Stability analysis ensures that the disease-free equilibrium point is locally asymptotically stable if carrying capacity greater than a certain threshold value.We have also derived the conditions for which endemic equilibrium is globally asymptotically stable that means the disease persists in the system. Numerical simulation suggests that the fear factor is an important role which is observed through Hopf-bifurcation.

scholarly journals Stability analysis of a dynamical model of tuberculosis with incomplete treatment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ihsan Ullah ◽  
Saeed Ahmad ◽  
Qasem Al-Mdallal ◽  
Zareen A. Khan ◽  
Hasib Khan ◽  
...  
Keyword(s):  
Disease Transmission ◽  
Contact Rate ◽  
Asymptotically Stable ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Efficient Treatment ◽  
Effective Contact ◽  
The Impact ◽  
Incomplete Treatment

Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ ℜ 0 ; if $\Re _{0}<1$ ℜ 0 < 1 , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}>1$ ℜ 0 > 1 , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

scholarly journals Mathematical Modelling and Analysis of Corruption of Morals amongst Adolescents with Control Measures in Kenya

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Nathan Oigo Mokaya ◽  
Haileyesus Tessema Alemmeh ◽  
Cyrus Gitonga Ngari ◽  
Grace Gakii Muthuri
Keyword(s):  
Integrated Control ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Control Measures ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Appropriate ◽  
Manifold Theory ◽  
Necessary And Sufficient ◽  
Well Posed

In the present paper, we formulate a new mathematical model for the dynamics of moral corruption with comprehensive age-appropriate sexual information and provision of guidance and counselling. The population is subdivided into three (3) different compartments according to their level of information on sexual matters. The model is proved to be both epidemiologically and mathematically well posed. The existence of unique morally corrupt-free and endemic equilibrium points is investigated. The basic reproduction number with respect to morally corrupt-free equilibrium is obtained using next generation matrix approach to monitor the dynamics of corrupt morals and ascertain its level in order to suggest effective intervention strategies to control this problem. The local as well as global asymptotic stability of these equilibrium points is studied. The analysis reveals a globally asymptotically stable morally corrupt-free equilibrium whenever ℛ 0 ≤ 1 and a globally asymptotically stable endemic equilibrium if otherwise. Further analysis, using center manifold theory, shows that the model exhibits forward bifurcation insinuating that the classical epidemiological requirement of ℛ 0 ≤ 1 is necessary and sufficient for elimination of moral corruption. A brief discussion on the graphical results using the available numerical procedures is shown. From numerical simulations, it was ascertain that integrated control strategy is the best approach to fight against moral corruption transmission. Lastly, some key parameters that show significance in the moral corruption elimination from the society are also exploited.

scholarly journals Global Stability of Malaria Transmission Dynamics Model with Logistic Growth

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Abadi Abay Gebremeskel
Keyword(s):  
Sensitivity Analysis ◽  
Global Stability ◽  
Malaria Transmission ◽  
Equilibrium Points ◽  
Transmission Dynamics ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Dynamics Model ◽  
Globally Asymptotically Stable ◽  
The Impact

Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0>1, then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.

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